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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.02732 |
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| _version_ | 1866915726891155456 |
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| author | Perrin, Derek Voloch, José Felipe |
| author_facet | Perrin, Derek Voloch, José Felipe |
| contents | We study $\ell$-isogeny graphs of ordinary elliptic curves defined over $\mathbb{F}_q$ with an added level structure. Given an integer $N$ coprime to $p$ and $\ell,$ we look at the graphs obtained by adding $Γ_0(N),$ $Γ_1(N),$ and $Γ(N)$-level structures to volcanoes. Given an order $\mathcal{O}$ in an imaginary quadratic field $K,$ we look at the action of generalised ideal class groups of $\mathcal{O}$ on the set of elliptic curves whose endomorphism rings are $\mathcal{O}$ along with a given level structure. We show how the structure of the craters of these graphs is determined by the choice of parameters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02732 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ordinary Isogeny Graphs with Level Structure Perrin, Derek Voloch, José Felipe Number Theory 14K02 (Primary), 11G20 (Secondary) We study $\ell$-isogeny graphs of ordinary elliptic curves defined over $\mathbb{F}_q$ with an added level structure. Given an integer $N$ coprime to $p$ and $\ell,$ we look at the graphs obtained by adding $Γ_0(N),$ $Γ_1(N),$ and $Γ(N)$-level structures to volcanoes. Given an order $\mathcal{O}$ in an imaginary quadratic field $K,$ we look at the action of generalised ideal class groups of $\mathcal{O}$ on the set of elliptic curves whose endomorphism rings are $\mathcal{O}$ along with a given level structure. We show how the structure of the craters of these graphs is determined by the choice of parameters. |
| title | Ordinary Isogeny Graphs with Level Structure |
| topic | Number Theory 14K02 (Primary), 11G20 (Secondary) |
| url | https://arxiv.org/abs/2411.02732 |